Teichmuller geometry of moduli space, I: Distance minimizing rays and the Deligne-Mumford compactification

Let $S$ be a closed, oriented surface with a finite (possibly empty) set of points removed. In this paper we relate two important but disparate topics in the study of the moduli space $\text{M}(S)$ of Riemann surfaces: Teichm"{u}ller geometry and the Deligne-Mumford compactification. We reconstruct the Deligne-Mumford compactification (as a metric stratified space) purely from the intrinsic metric geometry of $\text{M}(S)$ endowed with the Teichm"{u}ller metric. We do this by first classifying (globally) geodesic rays in $\text{M}(S)$ and determining precisely how pairs of rays asymptote. We construct an “iterated EDM ray space” functor, which is defined on a quite general class of metric spaces. We then prove that this functor applied to $\text{M}(S)$ produces the Deligne-Mumford compactification.